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flrv    _!_   £

Let z be replaced by-------. Then, for a denumerabk infinity

cz + d

of sets of values of a<b,c,d} there are uniform functions of 2, sav
F(z) is one of them, such that

+ *\

« + d)


Further, if a^bvc19dv and ajbzczdz are any two of the sets of
values of a.b&d, and if z be replaced first by -ilJ—*, and then,

in this, s be replaced by -2—!—-, giving, say, 1-1ZI—, then not

c22 -r ^2                  C^ -f Z>

only do we have

^feip-M = *"<=)• '•

but also

Further the set of all the substitutions

az + b

cz + d

(the arrow is read *is replaced by') which leave the value of
F(z) unchanged as just explained/om a group: the result of the
successive performance of two substitutions in the set,

is in the set; there is an 'identity substitution' in the set, namely
2—^2 (here a = 13 b = 0, c = Os d = 1); and finally each substi-
tution has a unique 'inverse* - that is, for each substitution in
the set there is a single other one which, if applied to the first,
will produce the identity substitution* In summary, using the
terminology of previous chapters, we see that F(z) is a function
which is invariant under an infinite group of linear fractional
transformations. Note that the infinity of substitutions is a